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#### Symplectic hypergeometric groups of degree six

##### External Resource

https://doi.org/10.1016/j.jalgebra.2021.02.010

(Publisher version)

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##### Fulltext (public)

2003.10191.pdf

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##### Citation

Bajpai, J., Dona, D., Singh, S., & Singh, S. V. (2021). Symplectic hypergeometric
groups of degree six.* Journal of Algebra,* *575*, 256-273.
doi:10.1016/j.jalgebra.2021.02.010.

Cite as: https://hdl.handle.net/21.11116/0000-000A-E65B-D

##### Abstract

Our computations show that there is a total of $40$ pairs of degree six

coprime polynomials $f,g$ where $f(x)=(x-1)^6$, $g$ is a product of cyclotomic

polynomials, $g(0)=1$ and $f,g$ form a primitive pair. The aim of this article

is to determine whether the corresponding $40$ symplectic hypergeometric groups

with a maximally unipotent monodromy follow the same dichotomy between

arithmeticity and thinness that holds for the $14$ symplectic hypergeometric

groups corresponding to the pairs of degree four polynomials $f,g$ where

$f(x)=(x-1)^4$ and $g$ is as described above. As a result we prove that at

least $18$ of these $40$ groups are arithmetic in $\mathrm{Sp}(6)$.

In addition, we extend our search to all degree six symplectic hypergeometric

groups. We find that there is a total of $458$ pairs of polynomials (up to

scalar shifts) corresponding to such groups. For $211$ of them, the absolute

values of the leading coefficients of the difference polynomials $f-g$ are at

most $2$ and the arithmeticity of the corresponding groups follows from Singh

and Venkataramana, while the arithmeticity of one more hypergeometric group

follows from Detinko, Flannery and Hulpke.

In this article, we show the arithmeticity of $160$ of the remaining $246$

hypergeometric groups.

coprime polynomials $f,g$ where $f(x)=(x-1)^6$, $g$ is a product of cyclotomic

polynomials, $g(0)=1$ and $f,g$ form a primitive pair. The aim of this article

is to determine whether the corresponding $40$ symplectic hypergeometric groups

with a maximally unipotent monodromy follow the same dichotomy between

arithmeticity and thinness that holds for the $14$ symplectic hypergeometric

groups corresponding to the pairs of degree four polynomials $f,g$ where

$f(x)=(x-1)^4$ and $g$ is as described above. As a result we prove that at

least $18$ of these $40$ groups are arithmetic in $\mathrm{Sp}(6)$.

In addition, we extend our search to all degree six symplectic hypergeometric

groups. We find that there is a total of $458$ pairs of polynomials (up to

scalar shifts) corresponding to such groups. For $211$ of them, the absolute

values of the leading coefficients of the difference polynomials $f-g$ are at

most $2$ and the arithmeticity of the corresponding groups follows from Singh

and Venkataramana, while the arithmeticity of one more hypergeometric group

follows from Detinko, Flannery and Hulpke.

In this article, we show the arithmeticity of $160$ of the remaining $246$

hypergeometric groups.